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Part I

Chapter 1 Introduction

1.1 Introductory remarks

The finite element method is now firmly accepted as a most powerful general technique for the numerical solution of a variety of problems encountered in engineering. Applications range from the stress analysis of solids to the solution of acoustical, neutron physics and fluid dynamics problems. Indeed the finite element process is now established as a general numerical method for the solution of partial differential equation systems, subject to known boundary and/or initial conditions.

For linear analysis, at least, the technique is widely employed as a design tool. Similar acceptance for nonlinear situations is dependent on two major factors. Firstly, in view of the increased numerical operations associated with nonlinear problems, considerable computing power is required. Developments in the last decade or so have ensured that high-speed digital computers which meet this need are now available and present indications are that reductions in unit computing costs will continue. Secondly, before the finite element method can be used in design, the accuracy of any proposed solution technique must be proven. The development of improved element characteristics and more efficient nonlinear solution algorithms and the experience gained in their application to engineering problems have ensured that nonlinear finite element analyses can now be performed with some confidence. Hence barriers to the common use of nonlinear finite element techniques are being rapidly removed and the process is already economically acceptable for selected industrial applications.

1.2 Aims and layout

The object of this book is to describe in detail the application of the finite element method to the solution of materially nonlinear engineering analysis problems. Unlike other texts on linear and nonlinear finite element analysis ^{(1-4)} which have dealt predominantly with theoretical aspects, this book is intended to be more practical and therefore focuses attention on the computer implementation of nonlinear finite element schemes.

Nonlinearities arise in engineering situations from several sources. For example a nonlinear material response can result from elasto-plastic material behaviour or from hyperelastic effects of some form. Additionally nonlinear

characteristics can be associated with temporal effects such as viscoplastic behaviour or dynamic transient phenomena. Each of these nonlinearities may occur in a variety of structural types such as two- or three-dimensional solids, frames, plates or shells. Therefore it becomes clear that a textbook dealing with nonlinear finite element programming must at least be restricted to selected topics. For this reason three classes of problems will be examined in depth in the three parts of this text.

Part I: One-dimensional materially nonlinear problems. All the essential features of a nonlinear finite element solution can be described in relation to one-dimensional models. The applications considered are:

● Nonlinear quasi-harmonic problems
● Nonlinear elastic situations

• Elasto-plastic behaviour of axial bar systems
  ● Time dependent elasto-viscoplastic analysis of bar systems
• Elasto-plastic beam bending

Part II: Two-dimensional materially nonlinear problems. In this part the ideas developed in Part I are extended to continuum problems. The following applications are presented:

• Elasto-plastic analysis of plane stress, plane strain and axisymmetric solids
  ● Time dependent elasto-viscoplastic analysis of plane stress, plane strain and axisymmetric solids
• Elasto-plastic plate bending problems

Part III: Nonlinear transient dynamic problems. In this time-dependent class of problems inertia effects are included in the analysis. In this part, the following topics are considered:

• Elasto-plastic and geometrically nonlinear material behaviour
  ● Explicit and implicit time integration schemes
  ● Combined explicit/implicit algorithms

It should be pointed out that several different programming options are open for solution of the above problems and the methods presented in this text are the ones which are physically the most clear and which experience indicates give reliable results for a wide range of applications. An important feature of this text is the step-by-step development of thirteen finite element programs to deal with the above problems.

For the one-dimensional applications considered in Part I, only a 2-node element with linear displacement variation between nodes is considered. This allows the basic steps of a nonlinear finite element analysis to be presented without unnecessary distractions. In Parts II and III of the text, where two-dimensional continuum and plate bending problems are considered, isoparametric elements are exclusively employed. In particular, a

4-node linear element and 8- and 9-node quadratic versions are used. These elements are illustrated in Fig. 1.1 and are extremely versatile, good performers which have been well tried and tested in both linear and nonlinear situations. A typical elasto-plastic application using 8-node isoparametric elements is shown in Fig. 1.2 where the incremental loading of a notched beam is illustrated. The progressive development of plastic zones with increasing load levels are compared for a Tresca and Von Mises yield criterion.

text_image

y x (a)

text_image

General node (b)

text_image

Central node (c)

Fig. 1.1 The two-dimensional isoparametric elements employed in the text: (a) Linear 4-node; (b) Serendipity 8-node; (c) Lagrangian 9-node.

The layout of the book will now be briefly described. The remainder of Chapter 1 discusses the basic notation and style adopted in program presentation.

Chapter 2 discusses the general nonlinear problem and some solution techniques are outlined. For the one-dimensional applications to be considered, basic theoretical expressions are developed in a form suitable for numerical solution.

In Chapter 3, the solution techniques presented in Chapter 2 are programmed in FORTRAN and numerical examples are solved for each separate application.

Chapter 4 is devoted to one-dimensional elasto-viscoplastic problems. The basic theory for this time-dependent phenomenon is first presented. The process is then coded and the program used to solve some numerical examples.

In Chapter 5 elasto-plastic beam bending is considered. This topic forms a bridge between uniaxial and continuum applications since now more than one degree of freedom exists at each nodal point. Some measure of continuum behaviour is also introduced since a layered approach is used to trace the development of plasticity through the cross-section of the beam.

other

Dimension	Value
Height (in mm)	0.5 IN
Width (in mm)	12.7 mm
Height (in mm)	0.75 IN
Width (in mm)	19.05 mm
Height (in mm)	6.0PY
Width (in mm)	5.5PY
Height (in mm)	6.0PY
Width (in mm)	0.010 IN
Height (in mm)	0.1667 IN
Width (in mm)	4.2 mm
Height (in mm)	0.3333 IN
P (Top)	22.2°

NOTCHED BEND SPECIMEN

\mathbf{P}_{\gamma} -Initial yield load for Von

Mises material =

536 lb (2.39 kN)

No strain hardening

Elastic modulus, E = 3 \times 10^{7} lb/in ^{2}

(2\times 10^{5}\mathrm{N / mm^{2}})

Poisson's ratio, \nu = 0.28

Yield stress, \sigma_{Y} = 6 \times 10^{4} \mathrm{~lb/in}^{2} = \frac{\mathrm{E}}{500}

ZONES OF

PLASTIC YIELD

AT VARIOUS

LOAD VALUES

contour

Label	Value
1.2Pγ	3.0
5.5	4.5
6.0	5.5
6.0	6.0

Fig. 1.2 Elasto-plastic analysis of a notched beam under bending showing plastic zone distributions for both a Von Mises and a Tresca yield criterion.

Chapter 6 forms an introduction to two-dimensional continuum problems. The basic theory for two-dimensional isoparametric elements is presented and some standard subroutines required for applications described in later chapters are listed. These include routines which perform some standard linear elastic operations, such as nodal load generation, equation solution, etc., as well as nonlinear subroutines common to more than one application.

Two-dimensional elasto-plastic problems are considered in Chapter 7. Basic theoretical expressions for a general continuum are first reviewed, and manipulated into forms convenient for numerical analysis. Particular expressions for plane stress/strain and axisymmetric situations are then developed and coded. Four different yield criteria are employed. The Tresca and Von Mises laws which closely approximate metal plasticity behaviour are considered and the Mohr–Coulomb and Drucker–Prager criteria, which are applicable to concrete, rocks and soil are presented.

Chapter 8 is concerned with the transient phenomenon of elastoviscoplasticity where again the situations of plane stress/strain and axial symmetry are considered. Both explicit and implicit time integration schemes are presented and the four yield criteria considered in Chapter 7 are employed. The FORTRAN program developed is illustrated by application to some numerical examples.

Elasto-plastic plate bending problems are discussed in Chapter 9. The basic theoretical expressions are presented in a form suitable for numerical analysis with both a layered and nonlayered approach to plastification through the plate thickness being considered. Treatment in this chapter is limited to the Tresca and Von Mises yield conditions.

Chapters 10 and 11 deal with the transient dynamic analysis of two-dimensional continua. In this application inertia effects are included in the computation and problems such as blast loading and seismic phenomena are considered. Nonlinear effects due to both elasto-plastic material behaviour and gross geometric deformations are included. Both explicit and implicit techniques are employed for the time integration of the equations of motion as well as a combined implicit/explicit algorithm. The computer codes developed are applied to the solution of some practical problems.

Finally in Chapter 12 further aspects of nonlinear material behaviour are discussed. Alternative solution techniques and material models are referred to and some additional fields of application indicated.

Three appendices are included which contain user instructions for the computer programs described throughout the text. Appendices I and II provide user instructions for one-dimensional and two-dimensional continuum problems respectively. A user's guide for transient dynamic problems is provided in Appendix III. Finally in Appendix IV sample input data and lineprinter output are provided for both one- and two-dimensional applications.

1.3 Program structure

1.3.1 Introduction

This section describes the main features of the computer programs to be developed later in the book. A modular approach is adopted, in that separate subroutines are employed to perform the various operations required in a nonlinear finite element analysis. Generally each program consists of 9 modules, each with a distinct operational function. Each module in turn is composed of one or more subroutines relevant only to its own needs and, in some cases, of subroutines which are common to several modules. Control of the modules is held by the main or master segment.

The modules, shown schematically in Fig. 1.3, are described in relation to their general functions as follows:

1. Initialisation or zeroing module—this is the first module entered and its function is to initialise to zero various vectors and matrices at the beginning of the solution process.
2. Data input and checking module—this is the second module entered. It handles input data defining the geometry, boundary conditions and material properties. This data is checked using diagnostic routines and if errors occur they are flagged and the remainder of the input data is printed out before the program is terminated. For isoparametric elements, Gaussian integration constants and mid-side nodal coordinates for straight-sided elements are also evaluated in this section. Once used this module is not needed again.
3. Loading module—this module organises the calculation of nodal forces due to the various forms of loading for two-dimensional application. These include pressure, gravity and concentrated loadings.
4. Load incrementing module—Any materially nonlinear finite element solution must proceed on an incremental basis. Therefore the function of this section is to control the incrementing of the applied loads evaluated by the loading module. It also ensures that any specified displacement values are also incrementally applied.
5. Stiffness module—this is the next module entered and organises the evaluation of the stiffness matrix for each element. The stiffness matrices are stored on disc and ordered in the sequence required for equation assembly and reduction.
6. Solution module—the general purpose of this routine is to assemble, reduce and solve the governing set of simultaneous equations to give the nodal displacements and force reactions at restrained nodal points.
7. Residual force module—the function of this module is to calculate the residual or 'out of balance' nodal forces at each stage of the analysis.
8. Convergence module—in this module the convergence of the nonlinear solution is checked against criteria given in later chapters.

9. Output module—this module organises the output of the requested quantities.

flowchart

graph TD
    A["Main or master segment"] --> B["Initialising or zeroing module"]
    A --> C["Data input and checking module"]
    A --> D["Loading module"]
    A --> E["Load incrementing module"]
    A --> F["Stiffness module"]
    A --> G["Solution module"]
    A --> H["Residual force module"]
    A --> I["Convergence module"]
    A --> J["Output module"]
    style A fill:#f9f,stroke:#333
    style B fill:#ccf,stroke:#333
    style C fill:#ccf,stroke:#333
    style D fill:#ccf,stroke:#333
    style E fill:#ccf,stroke:#333
    style F fill:#ccf,stroke:#333
    style G fill:#ccf,stroke:#333
    style H fill:#ccf,stroke:#333
    style I fill:#ccf,stroke:#333
    style J fill:#ccf,stroke:#333
    style F fill:#ccf,stroke:#333
    style G fill:#ccf,stroke:#333
    style H fill:#ccf,stroke:#333
    style I fill:#ccf,stroke:#333
    style J fill:#ccf,stroke:#333
    style A fill:#fff,stroke:#333
    style B fill:#fff,stroke:#333
    style C fill:#fff,stroke:#333
    style D fill:#fff,stroke:#333
    style E fill:#fff,stroke:#333
    style F fill:#fff,stroke:#333
    style G fill:#fff,stroke:#333
    style H fill:#fff,stroke:#333
    style I fill:#fff,stroke:#333
    style J fill:#fff,stroke:#333
    style A fill:#fff,stroke:#333
    style B fill:#fff,stroke:#333
    style C fill:#fff,stroke:#333
    style D fill:#fff,stroke:#333
    style E fill:#fff,stroke:#333
    style F fill:#fff,stroke:#333
    style G fill:#fff,stroke:#333
    note1["Increment loop"] --> A
    note2["Iteration loop"] --> A

Fig. 1.3 Program modules for nonlinear solution codes.

The main purpose of the main or master segment is to call the above modules and to control the load increments and iteration procedure according to the solution algorithm being employed and the convergence rate of the solution process.

1.3.2 Programming notation

In the programs presented in this text an attempt has been made to name variables in a logical manner. By choosing descriptive names, the use of many of the variables becomes self-apparent, thus assisting the reader in the task of program assimilation. All variable names are chosen to be 5 characters in length; this occasionally causes a little difficulty in abbreviation but has an advantage with regard to neatness of program presentation. For example, the following names will be employed.

NMATS The Number of different MATerialS PROPS ( ) The array of material PROPERTieS NEVAB The Number of Element VAriaBles NNODE The Number of NODEs per Element NDOFN The Number of Degrees Of Freedom per Node

Furthermore a ‘common root’ principle will be adopted; where a single basic variable name is employed with different prefixes depending on its usage in the program. In particular:

i) Prefix I, J or L will be used to indicate a DO loop variable
ii) Prefix K will indicate a counter
iii) Prefix M will indicate a maximum value
iv) Prefix N will indicate a given number

For example IPOIN, NPOIN, MPOIN will indicate respectively a particular nodal point, the number of nodal points in the problem and the maximum permissible number of nodal points in the program.

Similarly, any DO loop will be of the general form

KEVAB=0 DO 1 INODE=1, NNODE DO 1 IDOFN=1, NDOFN 1 KEVAB=KEVAB+1

which indicates that the outer and inner DO loop indices range respectively over the number of nodes per element and the number of degrees of freedom per node. The prefix K is employed in KEVAB to indicate a counter over the number of element variables, NEVAB.

All programming is undertaken in standard FORTRAN IV. A listing is presented for all subroutines described in this text and detailed notes on each group of statements are provided. Comment cards have also been used to assist in the understanding of the programs.